Finite Group Databasebeta

← #29669340

Subgroups — #29669340

Order 15616.

736 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456564C211yesmaximal, characteristic, complemented
2#21456560C212yesmaximal, complemented
3#21456193C212yesmaximal
4#21445481C212yesmaximal, complemented
5#21445480C212yesmaximal, complemented
6#21445478C212yesmaximal, complemented
7#21445430C212yesmaximal, complemented
8#21444122C211yesmaximal, characteristic, complemented
9C61×#3138C211yesmaximal, characteristic
10#16840319C2212yes—
11#16840160C2211yescharacteristic, complemented
12#16840158C2212yescomplemented
13#16840142C2211yescharacteristic, complemented
14#16840141C2212yescomplemented
15#16840126—24no—
16#16840114C2212yescomplemented
17#16840113C2212yescomplemented
18#16840112C2212yescomplemented
19#16840108C2212yes—
20#16840108C2212yes—
21#16840108C2212yes—
22#16840108C2212yes—
23#16840020—22no—
24#16840018—24no—
25#16839870—22no—
26#16839867—24no—
27#16839767—24no—
28C4×#11758194C2212yes—
29#16839484C2212yescomplemented
30#16839483C2211yescharacteristic, complemented
31C8×#92418C2211yescharacteristic, complemented
32C61×#533C2211yescharacteristic
33C2×#127×C61C2211yescharacteristic
34C61×#490—22no—
35C61×#486C2212yes—
36C61×#486C2212yes—
37C61×#418—22no—
38C4×M16×C61C2211yescharacteristic
39C2×Dic244—22no—
40C2×#11758202—24no—
41C2×#11758194D414yes—
42#14519777C2312yes—
43#14519777C2312yes—
44#14519769—22no—
45#14519756—22no—
46#14519756—22no—
47C4×Dic122—22no—
48C4×Dic122—22no—
49#14519736—22no—
50#14519709—22no—
51#14519708—24no—
52#14519702—24no—
53#14519701—24no—
54#14519700—24no—
55#14519698—24no—
56#14519698—24no—
57#14519698—24no—
58#14519698—24no—
59#14519688D414yes—
60C4×#92418C2312yes—
61C4×#92418C2312yes—
62C22×D4×C61—22no—
63C22×D4×C61—21no—
64C2×M16×C61D412yes—
65C4⋊Q8×C61C2312yes—
66#127×C61—22no—
67#127×C61—21no—
68#127×C61C2312yes—
69#125×C61—21no—
70C22⋊Q8×C61—22no—
71C4×Q8×C61—22no—
72#117×C61—21no—
73C2×C42×C61C2311yescharacteristic
74#102×C61—24no—
75#102×C61—24no—
76#97×C61D412yes—
77C4×C8×C61C2311yescharacteristic
78C4×C8×C61C2311yescharacteristic
79C24×C61—41no—
80C2×Q8×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—48no—
85C2×D4×C61—44no—
86C2×D4×C61—42no—
87C2×D4×C61—41no—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D411yescharacteristic
90C2×Dic122—22no—
91C2×Dic122—22no—
92M16×C61—42no—
93C2×C8×C61C2×D412yes—
94C2×C8×C61C2×D412yes—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—42no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C42×C61C2×D412yes—
101C42×C61C2×D411yescharacteristic
102C42×C61C2411yescharacteristic
103C22⋊Dic61—42no—
104#11758202—48no—
105C4⋊Dic61—22no—
106C4⋊Dic61—22no—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109#11758196—42no—
110#11758196—42no—
111C4×Dic61—42no—
112#11758194—44no—
113C2×#92418C2×D414yes—
114C2×#92418C2×D414yes—
115Dic244—44no—
116C23×C61—42no—
117C23×C61—84no—
118C23×C61—82no—
119C23×C61#12711yescharacteristic
120Q8×C61—44no—
121D4×C61—44no—
122D4×C61—44no—
123D4×C61—44no—
124D4×C61—84no—
125D4×C61—82no—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—42no—
129C2×C4×C61#12712yes—
130C2×C4×C61#12711yescharacteristic
131C2×C4×C61C22×D412yes—
132C2×C4×C61C22×D411yescharacteristic
133C2×Dic61—42no—
134C2×Dic61—42no—
135Dic122—44no—
136Dic122—44no—
137C8×C61—22no—
138C8×C61—22no—
139#92418—24no—
140#92418—24no—
141#19753—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61—82no—
147C22×C61—164no—
148C22×C61C2×#12711yescharacteristic
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—82no—
152C4×C61#57514yes—
153C4×C61#57512yes—
154Dic61—82no—
155Dic61—82no—
156#3185—611no—
157#3185—612no—
158#3138—612no—
159#3138D6111yescharacteristic
160#1720—612no—
161#1720—612no—
162#1720—612no—
163#1711—612no—
164#1455—611no—
165C2×C61—41no—
166C2×C61—162no—
167C2×C61#315312yes—
168C2×C61#315311yescharacteristic
169#533—612no—
170#533C2×D6111yescharacteristic
171C2×#127C2×D6111yescharacteristic
172#498—612no—
173#498—611no—
174#494—612no—
175#494—611no—
176#490—22no—
177#490—1224no—
178#487—612no—
179#487—612no—
180#487—612no—
181#486—612no—
182#486—612no—
183#486—612no—
184#486—612no—
185#486C2×D6112yes—
186#486C2×D6112yes—
187#470—1224no—
188#470—1222no—
189#441—1224no—
190#441—1222no—
191#418—22no—
192#418—1224no—
193C4×M16—612no—
194C4×M16C2×D6111yescharacteristic
195#322—611no—
196#322—612no—
197C82—611no—
198C61#1975311yescharacteristic, complemented, Sylow
199C22×D4—22no—
200C22×D4—21no—
201C2×Q16—1224no—
202C2×Q16—1222no—
203C2×M16—614no—
204C2×M16C22⋊D6112yes—
205C4⋊Q8—612no—
206C4⋊Q8—612no—
207C4⋊Q8C22×D6112yes—
208#127—22no—
209#127—21no—
210#127C22×D6112yes—
211#125—21no—
212#125—1222no—
213C22⋊Q8—22no—
214C22⋊Q8—1222no—
215C22⋊Q8—1222no—
216C4×Q8—22no—
217C4×Q8—1222no—
218C4×Q8—1222no—
219#117—21no—
220#117—1222no—
221C2×C42C22×D6111yescharacteristic
222#106—1224no—
223#106—1222no—
224#103—1224no—
225#103—1224no—
226#103—1224no—
227#102—24no—
228#102—24no—
229#102—1224no—
230#102—1224no—
231#102—1224no—
232#102—1224no—
233#97—614no—
234#97C22⋊D6112yes—
235C4×C8—612no—
236C4×C8—612no—
237C4×C8C22×D6111yescharacteristic
238C4×C8C22×D6111yescharacteristic
239C24—41no—
240C2×Q8—22no—
241C2×Q8—1222no—
242C2×Q8—1222no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4D4×D6112yes—
251C22×C4C2×C22⋊D6111yescharacteristic
252Q16—2444no—
253Q16—2448no—
254M16—42no—
255M16—2444no—
256C2×C8—614no—
257C2×C8—614no—
258C2×C8C2×C22⋊D6112yes—
259C2×C8C2×C22⋊D6112yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—1222no—
264C4⋊C4—1222no—
265C4⋊C4—1222no—
266C4⋊C4—1222no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C22⋊C4—41no—
270C22⋊C4—2442no—
271C42—41no—
272C42—2442no—
273C42D4×D6112yes—
274C42C2×C22⋊D6111yescharacteristic
275C42C23×D6111yescharacteristic
276C23—42no—
277C23—84no—
278C23—82no—
279C23#1451977311yescharacteristic
280Q8—44no—
281Q8—2444no—
282Q8—2444no—
283D4—44no—
284D4—44no—
285D4—44no—
286D4—84no—
287D4—82no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4#1451977311yescharacteristic
294C2×C4#1451977312yes—
295C2×C4C2×D4×D6112yes—
296C2×C4C22×C22⋊D6111yescharacteristic
297C8—22no—
298C8—22no—
299C8—1224no—
300C8—1224no—
301C22—22no—
302C22—21no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—164no—
307C22C2×#1451977311yescharacteristic
308C4—22no—
309C4—21no—
310C4—82no—
311C4—4882no—
312C4—4882no—
313C4#1684055814yes—
314C4#1684056012yes—
315C2—41no—
316C2—162no—
317C2#2145628112yes—
318C2#2145628311yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#127
prime radicalC2×#127×C61C22
hypercenter#3138D61
hypocenterC61#19753
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456564C2semidirect
2. #21456560C2semidirect
4. #21445481C2semidirect
5. #21445480C2semidirect
6. #21445478C2semidirect
7. #21445430C2semidirect
8. #21444122C2semidirect
11. #16840160C22semidirect
12. #16840158C22semidirect
13. #16840142C22semidirect
14. #16840141C22semidirect
16. #16840114C22semidirect
17. #16840113C22semidirect
18. #16840112C22semidirect
29. #16839484C22semidirect
30. #16839483C22semidirect
31. C8×#92418C22semidirect
198. C61#19753semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214565642
#214565602
#214561932
#214454812
#214454802
#214454782
#214454302
#214441222
C61×#31382
#1975361